Now consider constructing a navigation function from which a vector
field can be derived that satisfies constraints motivated by the
acceleration-based control model, (8.47). As usual, the
definition of a navigation function begins with the consideration of a
potential function,
. What properties does a
potential function need to have so that it may be considered as a
navigation function as defined in Section 8.4.1 and also
yield a vector field that satisfies an acceleration bound? Sufficient
conditions will be given that imply that a potential function will be
a navigation function that satisfies the bound.

To give the conditions, it will first be important to characterize
extrema of multivariate functions. Recall from basic calculus that a
function
has a critical point when the first
derivative is zero. At such points, the sign of the second derivative
indicates whether the critical point is a minimum or maximum. These
ideas can be generalized to higher dimensions. A critical
point of is one for which
. The Hessian of is defined as the
matrix

(8.48)

At each critical point, the Hessian gives some information about the
extremum. If the rank of at is , then the Hessian
indicates the kind of extremum. If (8.48) is positive
definite,8.13 then the achieves a local
minimum at . If (8.48) is negative
definite,8.14 then the achieves a local maximum at . In all other cases, is a saddle
point. If the rank of at is less than , then the
Hessian is degenerate. In this case the Hessian cannot classify
the type of extremum. An example of this occurs when lies in a
plateau (there is no direction in which increases or decreases.
Such behavior is obviously bad for a potential function because the
local operator would not be able to select a direction.

Suppose that the navigation function is required to be smooth, to
ensure the existence of a gradient at every point. This enables
gradient descent to be performed. If is
not contractible, then it turns out there must exist some critical
points other than at which
. The
critical points can even be used to infer the topology of , which
is the basic idea in the subject of Morse theory
[701,234]. Unfortunately, this implies that there does not
exist a solution navigation function for such spaces because the
definition in Section 8.4.1 required that the integral
curve from any state that can reach must reach it using the
vector field derived from the navigation function. If the initial
state is a critical point, the integral curve is constant (the state
remains at the critical point). Therefore, under the smoothness
constraint, the definition of a navigation function should be modified
to allow critical points at a small number of places (only on a set
that has measure zero). It is furthermore required that the set of
states from which the integral curves arrive at each critical point
(i.e., the domain of attraction of each critical point) has measure
zero. From all possible initial states, except from a set of measure
zero, the integral curves must reach , if it is reachable.
This is ensured in the following definition.

A function
is called a navigation
function in the sense of Rimon-Koditschek if [829]:

It is smooth (or at least ).

Among all values on the connected component of
that
contains , there is only one local minimum, which is at
.8.15

It is maximal and constant on
, the boundary of
.

It is a Morse function [701], which means that at
each critical point (i.e.,
), the Hessian of
is not degenerate.8.16 Such
functions are known to exist on any smooth manifold.

If is smooth in the sense, then by Sard's
Theorem [234] the set of critical points has measure zero.

Methods for constructing navigation functions are outlined in
[829] for a general family of problems in which
has
a semi-algebraic description. The basic idea is to start with simple
shapes over which a navigation function can be easily defined. One
example of this is a spherical subset of
, which contains
spherical obstacles. A set of distorting transformations is then
developed to adapt the navigation functions to other shapes while
ensuring that the four properties above are maintained. One such
transformation extends a ball into any visibility region (in the sense
defined in Section 8.4.3). This is achieved by smoothly
stretching out the ball into the shape of the visibility region.
(Such regions are sometimes called star-shaped.) The transformations given in
[829] can be combined to define navigation functions for a
large family of configuration spaces. The main problem is that the
configuration space obstacles and the connectivity of
are
represented only implicitly, which makes it difficult to correctly
apply the method to complicated high-dimensional problems. One of the
advantages of the approach is that proving convergence to the goal is
simplified. In many cases, Lyapunov stability analysis can be
performed (see Section 15.1.1).